A334570 -id:A334570 - cp's OEIS frontend

A334568 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j/j).

1, 1, -1, 1, -1, 1, 1, -1, 0, -1, 1, -1, 0, 2, 1, 1, -1, 0, 0, -2, -1, 1, -1, 0, 0, 6, -6, 1, 1, -1, 0, 0, 0, -6, 16, -1, 1, -1, 0, 0, 0, 24, -24, 20, 1, 1, -1, 0, 0, 0, 0, -24, -120, -132, -1, 1, -1, 0, 0, 0, 0, 120, -120, 540, -28, 1, 1, -1, 0, 0, 0, 0, 0, -120, -720, 1764, 1216, -1

Offset: 0

Seiichi Manyama, May 06 2020

			Square array begins:
   1,    1,    1,    1,    1,    1,    1, ...
  -1,   -1,   -1,   -1,   -1,   -1,   -1, ...
   1,    0,    0,    0,    0,    0,    0, ...
  -1,    2,    0,    0,    0,    0,    0, ...
   1,   -2,    6,    0,    0,    0,    0, ...
  -1,   -6,   -6,   24,    0,    0,    0, ...
   1,   16,  -24,  -24,  120,    0,    0, ...
  -1,   20, -120, -120, -120,  720,    0, ...
   1, -132,  540, -720, -720, -720, 5040, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..5 give A033999, A001464, A334569, A334570, A334571.

Cf. A330858, A334561.

A(0,k) = 1 and A(n,k) = - (n-1)! * Sum_{j=1..min(k,n)} A(n-j,k)/(n-j)!.

A334564 E.g.f.: exp(-(x + x^2 + x^3 + x^4)).

Original entry on oeis.org

1, -1, -1, -1, 1, 139, 151, -1429, -19039, -170281, 517231, 12144439, 108063649, 294730019, -12338255929, -161010756061, -834576525119, 10909293532591, 350211422028511, 3176429907556079, -8268487440910399, -816508001025525061, -14017265801126968841, -55997107943920795301

Offset: 0

Seiichi Manyama, May 06 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500

Column 4 of A334561.

Cf. A193930, A334570.

m = 25; Range[0, m]! * CoefficientList[Series[Exp[-(x + x^2 + x^3 + x^4)], {x, 0, m}], x] (* Amiram Eldar, May 03 2021 *)

N=40; x='x+O('x^N); Vec(serlaplace(exp(-x-x^2-x^3-x^4)))

a(0) = 1 and a(n) = - (n-1)! * Sum_{k=1..min(4,n)} k*a(n-k)/(n-k)!.

cp's OEIS Frontend

A334568 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j/j).

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